Problem: Simplify the following expression and state the condition under which the simplification is valid. $a = \dfrac{r^3 + 9r^2 - 10r}{-7r^2 + 700}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ a = \dfrac {r(r^2 + 9r - 10)} {-7(r^2 - 100)} $ $ a = -\dfrac{r}{7} \cdot \dfrac{r^2 + 9r - 10}{r^2 - 100} $ Next factor the numerator and denominator. $ a = - \dfrac{r}{7} \cdot \dfrac{(r + 10)(r - 1)}{(r + 10)(r - 10)}$ Assuming $r \neq -10$ , we can cancel the $r + 10$ $ a = - \dfrac{r}{7} \cdot \dfrac{r - 1}{r - 10}$ Therefore: $ a = \dfrac{ -r(r - 1)}{ 7(r - 10)}$, $r \neq -10$